Timelike electromagnetic kaon form factor

We compute the electromagnetic charged kaon form factor in the timelike region by employing a Poincaré covariant formalism of the Bethe-Salpeter equation to study quark-antiquark bound states in conjunction with the Schwinger-Dyson equation for the quark propagator. Following a recent kindred calculation of the timelike electromagnetic pion form factor, we include the most relevant intermediate composite particles permitted by their quantum numbers in the interaction kernel to allow for a decay mechanism for the resonances involved. This term augments the usual gluon mediated interaction between quarks. For a sufficiently low energy timelike probing photon, the electromagnetic form factor is saturated by the $\rho (770)$ and $\varphi (1020)$ resonances. We assume $SU(2)$ isospin symmetry throughout. Our results for the absolute value squared of the electromagnetic form factor agree qualitatively rather well and quantitatively moderately so with available experimental data.

Figure 1

IA diagrams representing the coupling of the external field to the valence quarks. In the case of the electromagnetic charged kaon form factor, the external field couples to the $u$ (left diagram) and $s$ (right diagram) quarks.

Figure 2

Beyond IA diagrams necessary for a consistent calculation when we include meson exchange into the interaction kernel. The left diagram corresponds to the IA diagram, the middle diagram represents the coupling of the photon to the meson and the last diagram the coupling of the photon with a meson-quark vertex. For this work, we have included only the first IA diagram.

Figure 3

Absolute value squared of the electromagnetic kaon form factor on the timelike $Q^2<0$ region. We use the parameters $\eta =1.6$ and $\mathrm{\Lambda }=0.74\mathrm{GeV}$. We provide a comparison with experimental measurements extracted from [13]. Both resonances ($\rho$ and $\varphi$) are reproduced in our calculation.

Figure 4

Absolute value squared of the timelike kaon form factor using different values for the free parameters in the MT interaction, the upper panel: $\eta =1.5$, $\mathrm{\Lambda }=0.78$ and the bottom panel: $\eta =1.6$, $\mathrm{\Lambda }=0.74$.

Figure 5

Truncations used herein for the BSE interaction kernel $K$ (lower diagram) and the quark SDE one (upper diagram). In the lower diagram, the terms on the right-hand side correspond to the rainbow-ladder, pion/kaon exchange, and the $s$- and the $u$-channel pion/kaon decay contributions to the truncation, respectively. The $s$- and the $u$-channel pion/kaon decay terms do not contribute to the quark SDE.